For questions regarding harmonic functions.

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3
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1answer
80 views

A harmonic function with sublinear growth at infinity is constant

Show that if $u$ is harmonic in $\mathbb R^n$ and $u=o(|x|)$, then $u$ is a constant.(Hint: use the solid version of the mean value property $u(x)=\frac{1}{\omega R^n} \int _{B_{R(x)}} u(y)dy$, and ...
2
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0answers
19 views

biharmonic complex analysis

Complex analysis texts typically discuss analytic functions whose real and imaginary components are harmonic and satisfy the Laplace equation, $\nabla^2 f = 0$. I am working with a complex function ...
1
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1answer
24 views

let $f(x)$ is real polynomial. Can we say that $f(\left| x \right|)$ is subharmonic?

A continuous function $\varphi :\mathbb{R} \to \mathbb{C}$ is subharmonic if and only if, for any closed disc in $U$ with centre $\lambda_0$ and radius $r$,$$\varphi ({\lambda _0}) \le \frac{1}{{2\pi ...
0
votes
1answer
25 views

Is the product of two subharmonic function necessarily subharmonic?

Defin: A continuous function $\varphi :\mathbb{R} \to \mathbb{C}$ is subharmonic if and only if, for any closed disc in $U$ with centre $\lambda_0$ and radius $r$,$$\varphi ({\lambda _0}) \le ...
0
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0answers
18 views

Prove $\lim |u(x,y)|e^{-|x|}=0$ for harmonic function $u$

Let $\epsilon >0$ and $\Omega \subset \{(x,y): -\frac{\pi}{2}+\epsilon<y<\frac{\pi}{2}-\epsilon\}$. Let $u$ be harmonic function in $\Omega$ such that $u=0$ on $\partial \Omega$. Prove ...
2
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0answers
39 views

Summation of series involving $\sinh$ of a square root

In a physics problem that I assigned in one of my classes recently, the following series arises: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + ...
0
votes
1answer
36 views

Square integrable harmonic function

Let $u$ be harmonic function on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} u^2 dx1...dxn < +\infty$ (1). I've got to prove that it implies $u=0$. My idea was to prove that (1) implies that $u$ ...
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0answers
19 views

Co-efficient Problem For Univalent harmonic functions on Unit disk

The Clunie Sheil Small conjecture for the second co-efficient of a univalent harmonic function on the unit disk is as follows:- Suppose, $h(z)+\overline{g(z)}$ is a one-one harmonic function on the ...
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0answers
50 views

Generalized Coupon Collector's Problem [duplicate]

I just read about the coupon collector's problem where you're trying to find out how many coupons you need to collect on average before you get one complete set. This turns out to be $nH(n)$. I was ...
1
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2answers
29 views

Derive mean value property of harmonic function using this method

We can view harmonic function $u(x,y,z)$ as a wave function that does not depend on time $t$. Let $\overline{u}(r)$ be its average value on the circle with radius $r$. Assume we have proved that it ...
1
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1answer
29 views

Laplace equation for a lens-shaped volume - any approximate analytical solutions?

I need to solve Laplace equation for a lens (drop on a surface) with constant boundary condition on the top surface and zero on the bottom surface (this is the simplest case, not the only one I ...
0
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0answers
42 views

Is this a right calculation of Fourier transform?

I am trying to solve: Calculate Fourier transform (on $-\infty < x < \infty $) of: $$ f(x) = \begin{cases} 1, & -L<x<L \\ 0, & |x|\geq L \end{cases} $$ My ...
0
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0answers
35 views

Why does solving the Laplace equation on a graph using the method of relaxation get different results than a spectral method?

When I use the method of relaxation to solve Laplace's equation on a graph where the boundary conditions are fixing a set of nodes to either 0 or 1 then the solution ends up being entirely between 0 ...
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0answers
9 views

Surface Integral of a harmonic function and mean value property

I want to find a general expression of the following integral, where $h$ is a harmonic function (we're in $\mathbb{R}^2$): $\int_{|x-y|\leq a^2}\frac{h(y)}{\sqrt{a^2-|x-y|^2}}dy$ I think I can ...
0
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0answers
21 views

Harmonic function on 3-dimensional lamplighter group

Can anyone give an example of a non-constant bounded harmonic function on the Lamplighter group $\mathbb{Z}_2 \wr \ \mathbb{Z}^3$? This function should exist, because the random walk on this group has ...
0
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0answers
24 views

Fourier transform on forced linear harmonic oscillator

I'm having trouble solving the following equation: $\ddot x + \omega^2 x = a \sin \Omega t$, where $t >0$, $\omega \neq \Omega$, $x(0+) = 0 = \dot x(0+)$. It is asked to be done by Fourier ...
0
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0answers
29 views

Function $\phi (x,y) \mapsto \phi (u,v)$ [ie:…] under the transformation $f(z) = u(x,y) + v(x,y)$ where $f(z)$ is analytic & $d_z \,f(z)$ is not $0$

I am just having a bit of trouble understanding what I am being asked. The ie: in title says $\phi [x(u,v), y(u,v)]$ Part a) is asking to do the laplacian, $\nabla^2_{x,y} \phi (x,y)$ and to write ...
-1
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0answers
22 views

is this a harmonic function

Is this a harmonic function? $\frac{\ln1}{(x-2)^{2}+(y+1)^{2}}$ I think this is a harmonic function, because it's equal to zero ($ln1$). But maybe I'm wrong? Thanks!
1
vote
1answer
50 views

Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that…

In reviewing complex analysis, I stumbled upon the following problem: Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that $g(x,y)=3$ when $x^2+y^2=1$ and $g(x,y)=8$ when ...
2
votes
1answer
39 views

Variational formulation of harmonicity on Riemannian manifolds

$\newcommand{\R}{\mathbb{R}}$ I am trying to follow a derivation of the first variation formula for the energy functional. (In "Selected Topics in Harmonic maps"). Here is the context: $M,N$ are ...
2
votes
1answer
21 views

Isometries of Riemannian manifolds are harmonic?

Let $(M,g),(N,h)$ be two Riemannian manifolds. Assume $f:M \to N$ is an isometric immersion. Is $f$ harmonic? (i.e a critical point of the energy functional) (I know this is true when ...
0
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2answers
21 views

Is this function harmonic?

$$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to figure out if this function is harmonic. I worked out all the algebra and my conclusion is no because the numerator of the final fraction is $2x^3+2xy^2$ ...
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0answers
44 views

How to classify harmonic functions on the punctured disk without the Schwartz reflection principle?

I am working through old qual problems at the University of Minnesota and am trying to find an alternate solution to the following problem. Determine all continuous functions on $\{ z : 0 < \left| ...
1
vote
1answer
33 views

Partial derivative of x - is quotient rule necessary?

Let $$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to determine if the given function is harmonic. I know that the 2nd partial derivative with respect to $x$ should, when added to the 2nd partial ...
0
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0answers
26 views

Find the harmonic conjugate of the following

I just want to make sure my reasoning is correct. I followed another similar question from this site. Also, is there a method I can use after coming to the conclusion below to check to make sure my ...
1
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0answers
35 views

How do I see if $g$ is a polynomial or not??

Let $u$ be a real valued harmonic function on $\mathbb{C}$. Let $g: \mathbb{R^2} \to \mathbb{R}$ be defined by: $$g(x,y)=\int_0^{2\pi}u(e^{i\theta}(x+iy))\sin \theta \,d\theta$$ Which of the ...
4
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0answers
59 views

Harmonic map into $S^n \times \mathbb{R}$

Consider a harmonic map $\Phi : \Sigma \to S^n \times \mathbb{R}$, where $\Sigma$ is a surface, and the metric on $S^n \times \mathbb{R}$ is given by the product metric. Choose local spherical (polar) ...
3
votes
1answer
37 views

A version of Casorati-Weierstrass for harmonic functions?

Suppose that $f:B(0,1)\setminus\left\{0\right\}\subset \mathbb{R}^n \to \mathbb{R}$ is a harmonic function. Clearly, the property that $\overline{f(B(0,\epsilon))}=\mathbb{R}$ for all $\epsilon>0$ ...
0
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0answers
45 views

Harmonic map and pullback metric

Let $\phi : M \to \mathbb{R}^n$ be a harmonic map, where $M$ is a Riemannian manifold. Let us take coordinates $(u_1, u_2,..., u_n)$ on $\mathbb{R}^n$ and express the Euclidean metric as $g = \Sigma ...
0
votes
2answers
25 views

Harmonic Motion - Fourier Approximation What does this mean below?

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ...
3
votes
1answer
60 views

Interior estimate for derivatives of harmonic function

I'm learning PDE from the book of Trudinger and Gilbarg and I'm attempting to prove the following theorem: Let $\hspace{0.1ex}u\hspace{0.1ex}$ be harmonic in $\hspace{0.1ex}\varOmega \subset ...
1
vote
1answer
36 views

Harmonic Conjugate (Multiplication, power, addition/subtraction)

The Question Suppose that $v$ is a harmonic conjugate for $u$ on a domain $D$. Prove that $u(x,y)^3 - 3u(x,y)v(x,y)^2$ is harmonic. I'm trying to prove that this function is also harmonic when $v$ ...
0
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0answers
29 views

Value of $f_x g_x+ f_y g_y + f_z g_z$ when $f$ and $g$ are harmonic

Let $f$ and $g$ be distinct real-valued harmonic functions, which not merely differ by a constant or are not merely multiples of each other. Also, assume that the first order partial derivatives of ...
3
votes
3answers
148 views

I need to solve $\phi (x,y) = \frac{2V}{\pi} \int_{0}^{\infty} \frac{\sin(kx)\cosh(ky) dk}{k\cosh(ka)}$

I start with a integral in complex plane $$\oint_c \frac{e^{izx} e^{zy} dz}{z\cosh(za)}$$ where $c$ is a countour starting in $z = -R$ along the real axis and jumping the pole at origin and continuing ...
0
votes
1answer
47 views

Laplace-Beltrami of the Gauss map

I'm looking for the proof of very nice identity about the Laplace-Beltrami operator of the Gauss map $N$ of a regular surface in $\mathbb{R}^3$ given by a patch $X$. I want to show that $$\Delta N = ...
0
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0answers
28 views

Laplacian on the sphere (proof of a proposition)

$$ Let \;f:U \to \mathbb{R}\;be\;a\;function\;on\;an\;open\;subset\;of\;S^{m-1}\;which\;is\;the\;restriction\;of\;a\;smooth\;function\;F:U{'} \to ...
0
votes
1answer
34 views

Find all the harmonics functions constants on the rays

I'm stuck with this exercise, I don't know how characterize the harmonic functions of the exercise. I'd appreciate your help. Thank you. Let $G=\mathbb C\setminus\{(-\infty,0]\}$. Find all the ...
0
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0answers
24 views

Non trivial boundaries for laplacian equation on rectangle

1) Can this Laplace equation, with its non trivial boundaries (on a rectangular domain), be solved analytically? $$\frac{d^2U}{dx^2}+\frac{d^2U}{dy^2}=0$$ $$U_x(0,y)=0\quad,\quad U_x(a,y)=f(y)$$ ...
2
votes
1answer
48 views

Dirichlet problem to the ball with boundary data $1-2y^2$.

Let $\omega=\{(x,y):x^2+y^2<1\}$ be the open unit disk in $\mathbb R^2$ with the boundary $\delta\omega$.If $u(x,y)$ be the solution of Dirichlet problem $$\begin{cases} u_{xx}+ u_{yy}=0 & ...
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0answers
65 views

Whittakers general solution to Laplace and its relation to separable variables

So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates $u=x+iy$ and $v=x-iy$. This can be extended to n dimensions as long as the complex ...
1
vote
1answer
47 views

Harmonic function zeros on open subset

Let $h$ be an harmonic function on $\Omega\subset\mathbb{C}$. Let $A\neq\emptyset$ an open subset of $\Omega$ such that $h\mid_A\equiv 0$. Prove $h\mid_\Omega\equiv 0$. I thought on taking a ...
0
votes
1answer
121 views

Prove the uniqueness of poisson equation with robin boundary condition

We have $\Delta u=f$ in $D$, and $\dfrac{\partial u}{\partial n}+au=h$ on boundary of D, where $D$ is a domain in three dimension and $a$ is a positive constant. $\dfrac{\partial u}{\partial ...
1
vote
1answer
59 views

The average of a subharmonic function on a circle increases with radius

Let $u$ be a subharmonic on open set $\Omega$. Let $a\in\Omega,R>0$ such that $B(a,r)\subset \Omega$. Prove $$v(\rho)=\int_0^{2\pi}u(a+\rho e^{it})dt$$ is a monotone increasing function on ...
0
votes
3answers
61 views

Prove a function is harmonic

This problem is from Conformal Mapping by Zeev Nehari: If $u(x,y)$ is harmonic and $r=(x^2+y^2)^{1/2}$, prove $u(xr^{-2}, yr^{-2})$ is harmonic. The hint is obvious: "Use polar coordinates." I ...
1
vote
1answer
33 views

Extending a harmonic function

Suppose that $u$ is a harmonic function on some open set $U$ (assume that $\overline{U}$ is compact). It is well known than in this case $u$ is smooth. Is it true that we can extend $u$ to the whole ...
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0answers
21 views

How to generalize a fact (convex function of a mtg is submtg) about martingales to multivalued martingales?

It's known that a convex function of a martingale is a submartingale. What about martingales with values in $\mathbb{R}^{n}$? Is is true that a subharmonic function of such a martingale is a ...
0
votes
2answers
41 views

Strong maxima and minima

I'm stuck with this problem, in particular at b): Let $u:D \subseteq \mathbb{R}^2 \to \mathbb{R} \in C^2$ a harmonic function. $u$ has a local maximum at point $\vec{p} \in D$. Then: (a) Show that, ...
2
votes
0answers
35 views

$\nabla^2 u = 0 $ and integral $u$ around $\partial B_\rho$

I'm doing exercises from book about vector calculus. And there is problem wich I'm not sure what to do. Let's see the hypotesis. Let, $D \subseteq \mathbb{R}^2$ an open set. Let, $u:\overline{D} \to ...
3
votes
0answers
38 views

Dirichlet energy of solution to Laplace equation

Suppose $V\subseteq\mathbb{R}^3$ is compact with a smooth boundary. I'm interested in the Dirichlet problem $\Delta u=0$ subject to boundary conditions $u|_{\partial V}=f$ for a given function ...
0
votes
0answers
38 views

Normal component of Laplace equation solution

Suppose $V\subset\mathbb{R}^3$ is a bounded region with a smooth boundary and that we are given $f:\partial V\rightarrow\mathbb R$. From classical PDE theory, the solution of the Laplace equation ...